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November  2016, 15(6): 2203-2219. doi: 10.3934/cpaa.2016034

Low regularity solutions for the (2+1)-dimensional Maxwell-Klein-Gordon equations in temporal gauge

Hartmut Pecher 1,

1. 

Fachbereich Mathematik und Naturwissenschaften, Bergische Universität Wuppertal, Gaußstr. 20, 42119 Wuppertal

Received  December 2015 Revised  July 2016 Published  September 2016

The Maxwell-Klein-Gordon equations in 2+1 dimensions in temporal gauge are locally well-posed for low regularity data even below energy level. The corresponding (3+1)-dimensional case was considered by Yuan. Fundamental for the proof is a partial null structure in the nonlinearity which allows to rely on bilinear estimates in wave-Sobolev spaces by d'Ancona, Foschi and Selberg, on an $(L^p_x L^q_t)$ - estimate for the solution of the wave equation, and on the proof of a related result for the Yang-Mills equations by Tao.
Keywords: local well-posedness, Maxwell-Klein-Gordon, temporal gauge..
Mathematics Subject Classification: 35Q40, 35L7.
Citation: Hartmut Pecher. Low regularity solutions for the (2+1)-dimensional Maxwell-Klein-Gordon equations in temporal gauge. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2203-2219. doi: 10.3934/cpaa.2016034
References:
[1]

P. d'Ancona, D. Foschi and S. Selberg, Product estimates for wave-Sobolev spaces in 2+1 and 1+1 dimensions, Contemporary Math., 526 (2010), 125-150. doi: 10.1090/conm/526/10379.  Google Scholar

[2]

M. Beals, Self-spreading of singularities for solutions to semilinear wave equations, Annals Math., 118 (1983), 187-214. doi: 10.2307/2006959.  Google Scholar

[3]

M. Czubak and N. Pikula, Low regularity well-posedness for the 2D Maxwell-Klein-Gordon equation in the Coulomb gauge, Comm. Pure Appl. Anal., 13 (2014), 1669-1683. doi: 10.3934/cpaa.2014.13.1669.  Google Scholar

[4]

S. Klainerman and M. Machedon (Appendices by J. Bougain and D. Tataru), Remark on Strichartz-type inequalities, Int. Math. Res. Notices, 5 (1996), 201-220. doi: 10.1155/S1073792896000153.  Google Scholar

[5]

S. Klainerman and M. Machedon, On the Maxwell-Klein-Gordon equation with finite energy, Duke Math. J., 74 (1994), 19-44. doi: 10.1215/S0012-7094-94-07402-4.  Google Scholar

[6]

M. Machedon and J. Sterbenz, Almost optimal local well-posedness for the (3+1)-dimensional Maxwell-Klein-Gordon equations, J. AMS, 17 (2004), 297-359. doi: 10.1090/S0894-0347-03-00445-4.  Google Scholar

[7]

V. Moncrief, Global existence of Maxwell-Klein-Gordon fields in (2+1)-dimensional spacetime, J. Math. Phys., 21 (1980), 2291-2296. doi: 10.1063/1.524669.  Google Scholar

[8]

H. Pecher, Low regularity local well-posedness for the Maxwell-Klein-Gordon equations in Lorenz gauge, Adv. Diff. Equ., 19 (2014), 359-386.  Google Scholar

[9]

H. Pecher, Unconditional global well-posedness in energy space for the Maxwell-Klein-Gordon system in temporal gauge, Adv. Diff. Equ., 20 (2015), 1009-1032.  Google Scholar

[10]

S. Selberg, Multilinear Space-time Estimates and Applications to Local Exisatence Theory for Nonlinear Wave Equations, PhD. Thesis Princeton, 1999.  Google Scholar

[11]

S. Selberg, Anisotropic bilinear $L^2$ estimates related to the 3D wave equation, Int. Math. Res. Not., 2008, art. ID rnn107.  Google Scholar

[12]

S. Selberg and A. Tesfahun, Finite energy global well-posedness of the Maxwell-Klein-Gordon system in Lorenz gauge, Comm. PDE, 35 (2010), 1029-1057. doi: 10.1080/03605301003717100.  Google Scholar

[13]

T. Tao, Multilinear weighted convolutions of $L^2$-functions and applications to non-linear dispersive equations, Amer. J. Math., 123 (2001), 838-908.  Google Scholar

[14]

T. Tao, Local well-posedness of the Yang-Mills equation in the temporal gauge below the energy norm, J. Diff. Equ., 189 (2003), 366-382. doi: 10.1016/S0022-0396(02)00177-8.  Google Scholar

[15]

J. Yuan, Global solutions of two coupled Maxwell systems in the temporal gauge, Discr. Cont. Dyn. Syst., 36 (2016), 1709-1719. doi: 10.3934/dcds.2016.36.1709.  Google Scholar

show all references

References:
[1]

P. d'Ancona, D. Foschi and S. Selberg, Product estimates for wave-Sobolev spaces in 2+1 and 1+1 dimensions, Contemporary Math., 526 (2010), 125-150. doi: 10.1090/conm/526/10379.  Google Scholar

[2]

M. Beals, Self-spreading of singularities for solutions to semilinear wave equations, Annals Math., 118 (1983), 187-214. doi: 10.2307/2006959.  Google Scholar

[3]

M. Czubak and N. Pikula, Low regularity well-posedness for the 2D Maxwell-Klein-Gordon equation in the Coulomb gauge, Comm. Pure Appl. Anal., 13 (2014), 1669-1683. doi: 10.3934/cpaa.2014.13.1669.  Google Scholar

[4]

S. Klainerman and M. Machedon (Appendices by J. Bougain and D. Tataru), Remark on Strichartz-type inequalities, Int. Math. Res. Notices, 5 (1996), 201-220. doi: 10.1155/S1073792896000153.  Google Scholar

[5]

S. Klainerman and M. Machedon, On the Maxwell-Klein-Gordon equation with finite energy, Duke Math. J., 74 (1994), 19-44. doi: 10.1215/S0012-7094-94-07402-4.  Google Scholar

[6]

M. Machedon and J. Sterbenz, Almost optimal local well-posedness for the (3+1)-dimensional Maxwell-Klein-Gordon equations, J. AMS, 17 (2004), 297-359. doi: 10.1090/S0894-0347-03-00445-4.  Google Scholar

[7]

V. Moncrief, Global existence of Maxwell-Klein-Gordon fields in (2+1)-dimensional spacetime, J. Math. Phys., 21 (1980), 2291-2296. doi: 10.1063/1.524669.  Google Scholar

[8]

H. Pecher, Low regularity local well-posedness for the Maxwell-Klein-Gordon equations in Lorenz gauge, Adv. Diff. Equ., 19 (2014), 359-386.  Google Scholar

[9]

H. Pecher, Unconditional global well-posedness in energy space for the Maxwell-Klein-Gordon system in temporal gauge, Adv. Diff. Equ., 20 (2015), 1009-1032.  Google Scholar

[10]

S. Selberg, Multilinear Space-time Estimates and Applications to Local Exisatence Theory for Nonlinear Wave Equations, PhD. Thesis Princeton, 1999.  Google Scholar

[11]

S. Selberg, Anisotropic bilinear $L^2$ estimates related to the 3D wave equation, Int. Math. Res. Not., 2008, art. ID rnn107.  Google Scholar

[12]

S. Selberg and A. Tesfahun, Finite energy global well-posedness of the Maxwell-Klein-Gordon system in Lorenz gauge, Comm. PDE, 35 (2010), 1029-1057. doi: 10.1080/03605301003717100.  Google Scholar

[13]

T. Tao, Multilinear weighted convolutions of $L^2$-functions and applications to non-linear dispersive equations, Amer. J. Math., 123 (2001), 838-908.  Google Scholar

[14]

T. Tao, Local well-posedness of the Yang-Mills equation in the temporal gauge below the energy norm, J. Diff. Equ., 189 (2003), 366-382. doi: 10.1016/S0022-0396(02)00177-8.  Google Scholar

[15]

J. Yuan, Global solutions of two coupled Maxwell systems in the temporal gauge, Discr. Cont. Dyn. Syst., 36 (2016), 1709-1719. doi: 10.3934/dcds.2016.36.1709.  Google Scholar

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